Modern neural networks often have great expressive power and can be trained to overfit the training data, while still achieving a good test performance. This phenomenon is referred to as "benign overfitting". Recently, there emerges a line of works studying "benign overfitting" from the theoretical perspective. However, they are limited to linear models or kernel/random feature models, and there is still a lack of theoretical understanding about when and how benign overfitting occurs in neural networks. In this paper, we study the benign overfitting phenomenon in training a two-layer convolutional neural network (CNN). We show that when the signal-to-noise ratio satisfies a certain condition, a two-layer CNN trained by gradient descent can achieve arbitrarily small training and test loss. On the other hand, when this condition does not hold, overfitting becomes harmful and the obtained CNN can only achieve constant level test loss. These together demonstrate a sharp phase transition between benign overfitting and harmful overfitting, driven by the signal-to-noise ratio. To the best of our knowledge, this is the first work that precisely characterizes the conditions under which benign overfitting can occur in training convolutional neural networks.
相關內容
Graph Convolutional Networks (GCNs) are one of the most popular architectures that are used to solve classification problems accompanied by graphical information. We present a rigorous theoretical understanding of the effects of graph convolutions in multi-layer networks. We study these effects through the node classification problem of a non-linearly separable Gaussian mixture model coupled with a stochastic block model. First, we show that a single graph convolution expands the regime of the distance between the means where multi-layer networks can classify the data by a factor of at least $1/\sqrt[4]{\mathbb{E}{\rm deg}}$, where $\mathbb{E}{\rm deg}$ denotes the expected degree of a node. Second, we show that with a slightly stronger graph density, two graph convolutions improve this factor to at least $1/\sqrt[4]{n}$, where $n$ is the number of nodes in the graph. Finally, we provide both theoretical and empirical insights into the performance of graph convolutions placed in different combinations among the layers of a network, concluding that the performance is mutually similar for all combinations of the placement. We present extensive experiments on both synthetic and real-world data that illustrate our results.
Convolutional neural network (CNN) has achieved impressive success in computer vision during the past few decades. As the core of CNNs, the image convolution operation helps CNNs to get good performance on image-related tasks. However, the image convolution is hard to be implemented and parallelized. This paper proposes a novel neural network model, namely CEMNet, which can be trained in the frequency domain. The most important motivation of this research is that we can use the straightforward element-wise multiplication operation to replace the image convolution in the frequency domain based on the Cross-Correlation Theorem. We further introduce a Weight Fixation mechanism to alleviate the problem of over-fitting, and analyze the working behavior of Batch Normalization, Leaky ReLU, and Dropout in the frequency domain to design their counterparts for CEMNet. Also, to deal with complex inputs brought by Discrete Fourier Transform, we design a two-branches network structure for CEMNet. Experimental results imply that CEMNet achieves good performance on MNIST and CIFAR-10 databases. To the best of our knowledge, CEMNet is the first model trained in Fourier Domain that achieves more than 70\% validation accuracy on CIFAR-10 database.
Momentum methods, including heavy-ball~(HB) and Nesterov's accelerated gradient~(NAG), are widely used in training neural networks for their fast convergence. However, there is a lack of theoretical guarantees for their convergence and acceleration since the optimization landscape of the neural network is non-convex. Nowadays, some works make progress towards understanding the convergence of momentum methods in an over-parameterized regime, where the number of the parameters exceeds that of the training instances. Nonetheless, current results mainly focus on the two-layer neural network, which are far from explaining the remarkable success of the momentum methods in training deep neural networks. Motivated by this, we investigate the convergence of NAG with constant learning rate and momentum parameter in training two architectures of deep linear networks: deep fully-connected linear neural networks and deep linear ResNets. Based on the over-parameterization regime, we first analyze the residual dynamics induced by the training trajectory of NAG for a deep fully-connected linear neural network under the random Gaussian initialization. Our results show that NAG can converge to the global minimum at a $(1 - \mathcal{O}(1/\sqrt{\kappa}))^t$ rate, where $t$ is the iteration number and $\kappa > 1$ is a constant depending on the condition number of the feature matrix. Compared to the $(1 - \mathcal{O}(1/{\kappa}))^t$ rate of GD, NAG achieves an acceleration over GD. To the best of our knowledge, this is the first theoretical guarantee for the convergence of NAG to the global minimum in training deep neural networks. Furthermore, we extend our analysis to deep linear ResNets and derive a similar convergence result.
Due to their increasing spread, confidence in neural network predictions became more and more important. However, basic neural networks do not deliver certainty estimates or suffer from over or under confidence. Many researchers have been working on understanding and quantifying uncertainty in a neural network's prediction. As a result, different types and sources of uncertainty have been identified and a variety of approaches to measure and quantify uncertainty in neural networks have been proposed. This work gives a comprehensive overview of uncertainty estimation in neural networks, reviews recent advances in the field, highlights current challenges, and identifies potential research opportunities. It is intended to give anyone interested in uncertainty estimation in neural networks a broad overview and introduction, without presupposing prior knowledge in this field. A comprehensive introduction to the most crucial sources of uncertainty is given and their separation into reducible model uncertainty and not reducible data uncertainty is presented. The modeling of these uncertainties based on deterministic neural networks, Bayesian neural networks, ensemble of neural networks, and test-time data augmentation approaches is introduced and different branches of these fields as well as the latest developments are discussed. For a practical application, we discuss different measures of uncertainty, approaches for the calibration of neural networks and give an overview of existing baselines and implementations. Different examples from the wide spectrum of challenges in different fields give an idea of the needs and challenges regarding uncertainties in practical applications. Additionally, the practical limitations of current methods for mission- and safety-critical real world applications are discussed and an outlook on the next steps towards a broader usage of such methods is given.
Deep graph neural networks (GNNs) have achieved excellent results on various tasks on increasingly large graph datasets with millions of nodes and edges. However, memory complexity has become a major obstacle when training deep GNNs for practical applications due to the immense number of nodes, edges, and intermediate activations. To improve the scalability of GNNs, prior works propose smart graph sampling or partitioning strategies to train GNNs with a smaller set of nodes or sub-graphs. In this work, we study reversible connections, group convolutions, weight tying, and equilibrium models to advance the memory and parameter efficiency of GNNs. We find that reversible connections in combination with deep network architectures enable the training of overparameterized GNNs that significantly outperform existing methods on multiple datasets. Our models RevGNN-Deep (1001 layers with 80 channels each) and RevGNN-Wide (448 layers with 224 channels each) were both trained on a single commodity GPU and achieve an ROC-AUC of $87.74 \pm 0.13$ and $88.14 \pm 0.15$ on the ogbn-proteins dataset. To the best of our knowledge, RevGNN-Deep is the deepest GNN in the literature by one order of magnitude. Please visit our project website //www.deepgcns.org/arch/gnn1000 for more information.
Deep learning methods for graphs achieve remarkable performance on many node-level and graph-level prediction tasks. However, despite the proliferation of the methods and their success, prevailing Graph Neural Networks (GNNs) neglect subgraphs, rendering subgraph prediction tasks challenging to tackle in many impactful applications. Further, subgraph prediction tasks present several unique challenges, because subgraphs can have non-trivial internal topology, but also carry a notion of position and external connectivity information relative to the underlying graph in which they exist. Here, we introduce SUB-GNN, a subgraph neural network to learn disentangled subgraph representations. In particular, we propose a novel subgraph routing mechanism that propagates neural messages between the subgraph's components and randomly sampled anchor patches from the underlying graph, yielding highly accurate subgraph representations. SUB-GNN specifies three channels, each designed to capture a distinct aspect of subgraph structure, and we provide empirical evidence that the channels encode their intended properties. We design a series of new synthetic and real-world subgraph datasets. Empirical results for subgraph classification on eight datasets show that SUB-GNN achieves considerable performance gains, outperforming strong baseline methods, including node-level and graph-level GNNs, by 12.4% over the strongest baseline. SUB-GNN performs exceptionally well on challenging biomedical datasets when subgraphs have complex topology and even comprise multiple disconnected components.
Graph Convolutional Networks (GCNs) and their variants have experienced significant attention and have become the de facto methods for learning graph representations. GCNs derive inspiration primarily from recent deep learning approaches, and as a result, may inherit unnecessary complexity and redundant computation. In this paper, we reduce this excess complexity through successively removing nonlinearities and collapsing weight matrices between consecutive layers. We theoretically analyze the resulting linear model and show that it corresponds to a fixed low-pass filter followed by a linear classifier. Notably, our experimental evaluation demonstrates that these simplifications do not negatively impact accuracy in many downstream applications. Moreover, the resulting model scales to larger datasets, is naturally interpretable, and yields up to two orders of magnitude speedup over FastGCN.
Deep Convolutional Neural Networks (CNNs) are a special type of Neural Networks, which have shown state-of-the-art results on various competitive benchmarks. The powerful learning ability of deep CNN is largely achieved with the use of multiple non-linear feature extraction stages that can automatically learn hierarchical representation from the data. Availability of a large amount of data and improvements in the hardware processing units have accelerated the research in CNNs and recently very interesting deep CNN architectures are reported. The recent race in deep CNN architectures for achieving high performance on the challenging benchmarks has shown that the innovative architectural ideas, as well as parameter optimization, can improve the CNN performance on various vision-related tasks. In this regard, different ideas in the CNN design have been explored such as use of different activation and loss functions, parameter optimization, regularization, and restructuring of processing units. However, the major improvement in representational capacity is achieved by the restructuring of the processing units. Especially, the idea of using a block as a structural unit instead of a layer is gaining substantial appreciation. This survey thus focuses on the intrinsic taxonomy present in the recently reported CNN architectures and consequently, classifies the recent innovations in CNN architectures into seven different categories. These seven categories are based on spatial exploitation, depth, multi-path, width, feature map exploitation, channel boosting and attention. Additionally, it covers the elementary understanding of the CNN components and sheds light on the current challenges and applications of CNNs.
We propose a Bayesian convolutional neural network built upon Bayes by Backprop and elaborate how this known method can serve as the fundamental construct of our novel, reliable variational inference method for convolutional neural networks. First, we show how Bayes by Backprop can be applied to convolutional layers where weights in filters have probability distributions instead of point-estimates; and second, how our proposed framework leads with various network architectures to performances comparable to convolutional neural networks with point-estimates weights. In the past, Bayes by Backprop has been successfully utilised in feedforward and recurrent neural networks, but not in convolutional ones. This work symbolises the extension of the group of Bayesian neural networks which encompasses all three aforementioned types of network architectures now.
小貼士
登錄享
相關主題
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191